Constrained topological degree and positive solutions of fully nonlinear boundary value problems

In the first part of the paper we provide a construction of an abstract homotopy invariant detecting zeros of maps of the form −A+F where is a densely defined m-accretive operator in a Banach space E and is a tangent field defined on an open subset U of a neighborhood retract M being invariant with respect to the resolvents of A. The construction is performed under the assumption that resolvents of A are completely continuous. In the second part we derive index formulae for isolated zeros and apply them to show the existence of nontrivial positive steady state solutions for a class of nonlinear reaction-diffusion equations and equations with one-dimensional p-Laplacian with possibly non-positive perturbations as well as some controlled Neumann-like problems.